Equivariant Hodge-Deligne polynomials of symmetric products of algebraic groups

Let X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products SymnX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Sym}^{n}X$$\end{document} when the cohomology of X is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials μXnSnt,u,v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{X^{n}}^{S_{n}}\left( t,u,v\right) $$\end{document}, codifying the permutation action of Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n}$$\end{document} as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products μSymnXt,u,v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\mathrm {Sym}^{n}X}\left( t,u,v\right) $$\end{document}. These formulas are then applied to the case of linear algebraic groups.